_07A_ELC4340_Spring13_Transformers.doc
Transformers
Transformers. Transformer phase shift. Wye-delta connections and impact on zero sequence.
Inductance and capacitance calculations for transmission lines. GMR, GMD, L, and C matrices,
effect of ground conductivity. Underground cables.
Equivalent Circuits
The standard transformer equivalent circuit used in power system simulation is shown below,
where the R and X terms represent the series resistance and leakage reactance, and N1 and N2
represent the transformer turns. Note that the shunt terms are usually ignored in the model..
R
jX
N1
N2
Figure 1. Power System Model for Transformer
Three-phase transformers can consist of either three separate single-phase transformers, or three
windings on a three-legged, four-legged, or five-legged core. The high-voltage and low-voltage
sides can be connected independently in either wye or delta.
High-Voltage Side
A
B
C
Low-Voltage Side
Figure 2. A Three-Phase Ground-Wye Grounded-Wye Transformer
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High-Voltage Side
A
B
C
Low-Voltage Side
Figure 3. A Three-Phase Delta-Delta Transformer
The transformer impedances consist of winding resistances and leakage reactances. There are no
mutual resistances, and the mutual leakage reactances between the separate phase a-b-c coils are
negligible. Hence, in symmetrical components, S = R + jX, and M = 0, so that S + 2M = S - M =
R + jX, so therefore the positive and negative sequence impedances of a transformer are
Z1  Z 2  R  jX .
One must remember that no zero sequence currents can flow into a three-wire connection.
Therefore, the zero sequence impedance of a transformer depends on the winding connections.
In the case where one side of a transformer is connected grounded-wye, and the other side is
delta, circulating zero sequence currents can be induced in the delta winding. In that case, the
zero sequence impedance "looking into" the transformer is different on the two sides.
The zero sequence equivalent circuits for three-phase transformers is given in Figure 4.
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R + jX
Grounded Wye - Grounded Wye
Grounded Wye - Delta
R + jX
Grounded Wye - Ungrounded Wye
R + jX
Ungrounded Wye - Delta
R + jX
Delta - Delta
R + jX
Figure 4. Zero Sequence Impedance Equivalent Circuits for Three-Phase Transformers
A wye-delta transformer connection introduces a 30o phase shift in positive/negative sequence
voltages and currents because of the relative shift between line-to-neutral and line-to-ground
voltages. Transformers are labeled so that
1. High side positive sequence voltages and currents lead those on the low side by 30o.
2. High side negative sequence voltages and currents lag those on the low side by 30o.
3. There is no phase shift for zero sequence.
Transformer tap magnitudes can be adjusted to control voltage, and transformer phase shifts can
be adjusted to control active power flow. The effect of these "off-nominal" adjustments can be
incorporated into a pi-equivalent circuit model for a transformer.
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Bus i
Bus k' y
t / :1
Bus k
Ik --->
Ii --->
Figure 5. Off-Nominal Transformer Circuit Model
Assume that the transformer in Figure 5 has complex "off-nominal" tap t t and series
admittance y. The relationship between the voltage on opposite sides of the transformer tap is
~
Vi
~
Vk ' 
, and since the power on both sides of the ideal transformer must be the same, then
t t
~~
~ ~
~
~
Vi I i*  Vk ' I k*' , implying that I k '  I i t  t . Now, suppose that the transformer can be
modeled by the following pi-equivalent circuit of Figure 6:
Bus i
Ii --->
Bus k
yik
<--- -Ik
ykk
yii
Figure 6. Pi-Equivalent Model of Transformer
Admittances yii, yik, and ykk can be found so that the above circuit is equivalent to Figure 5. This
can be accomplished by forcing the terminal behavior to be the same. For the above circuit, the
appropriate equations are




~
~ ~
~
~
~ ~
~
I i  Vi  Vk yik  Vi yii , and  I k  Vk  Vi yik  Vk ykk ,
or in matrix form
~
 I i   yii  yik
 ~ 
 I k    yik
~
  Vi 
~  .
y kk  yik  Vk 
 yik
For Figure 5, the terminal equations are
~
 Vi
~
~
~
~ 

I k  Vk '  Vk y  
 Vk  y ,
 t t



and since
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~
Ii 
~
Ik
,
t   t
then
~
~
Vi
Vk
~ 
Ii  

 t t  t   t t   t

y .


In matrix form,
y 
~
t   t  Vi 
 ~  .
y  Vk 

y

~

 Ii 
t t  t   t
 ~ 
y
 I k  

t t
Comparing the two sets of terminal equations shows that equality can be reached if the shunt
branch in the equivalent circuit, yik, can have two values:
y
from the perspective of Kirchhoff's current law at bus i,
t   t
from the perspective of Kirchhoff's current law at bus k.
yik 
Note that if the tap does not include an off-nominal phase shift, then yik 
yik 
y
t t
y
from either
t
direction.
Next, solving for yii and ykk yields
yii 
y
y
y


t t  t   t t   t t   t
y kk  y 

y
1
 y1 
t t
 t t
 1


 1 ,
 t t


 .

Neutral Grounding Impedance
If the wye-side of a transformer or wye-connected load is grounded through a grounding
impedance Zg, the grounding impedance is "invisible" to the positive and negative sequence
currents since their corresponding voltages at the wye-point is always zero due to symmetry.
However, since the neutral current is three-times the zero sequence current, the voltage drop on
the grounding impedance is 3Iao. For that reason, the zero sequence equivalent circuit for a
grounding impedance must contain 3Zg.
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Iao
Z
Iao
Z
Iao
Z
+
Zg
Za1 = Za2 = Z
Zao = Z + 3Zg
Vao = 3 Iao Zg
-
Figure 7. Effect of Grounding Impedance on Sequence Impedances
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